14.4 Extrema on a Curve in Three Dimensions

A curve
C
in three dimensions can be defined by two equations (that is as the intersection of two surfaces) or by use of a single parameter as in two dimensions.

If
q
is an extreme values of
F
on
C
we cannot have
∇&LongRightArrow;F·t&LongRightArrow;
non-zero at argument
q
, by our general principle; otherwise
F
will be larger on one side of
q
and smaller on the other than its value at
q
on
C
.

The implications of this condition are different here however. We can no longer say that
∇&LongRightArrow;F
points in some particular direction at an extremal point. Rather it must be normal to some particular direction, that of the tangent vector to
C
at such points.

When
C
is described by two equations,
G=0
and
H=0,t&LongRightArrow;
is in the direction of
∇&LongRightArrow;G×∇&LongRightArrow;H
, and the statement that
∇&LongRightArrow;F
has no component in that direction is the statement that
∇&LongRightArrow;Flies in the plane of
∇&LongRightArrow;G and
∇&LongRightArrow;H
and so the volume of their parallelepiped is 0 and
the determinant whose
columns are all these grads must be 0.

This condition and
G=0
and
H=0
determine
x,y
and
z
at critical points.

Another way to state the same condition is to use two Lagrange Multipliers, say
c
and
d
and write
∇&LongRightArrow;F=c∇&LongRightArrow;G+d∇&LongRightArrow;H
. We can solve the three equations obtained by writing all three components of this vector equation and use them and
G=0
and
H=0
, to solve for
c,d,x,y
, and
z
.

Exercises:

14.6 Given a curve defined as the intersection of the surfaces defined by equations
xyz=1, and
x2+2y2+3z2=7, find equations determining the critical
points of
2x3−y3 by the determinantal approach.

14.7 Write the equations for the critical points obtained using the Lagrange
Multipliers approach for the same problem.

14.8 We seek the critical points for
F on the curve
x=5sin⁡t,y=3cos⁡3t,z=sin⁡2t, for
t=0 to
2π, with
F=x2+y2+z2. Write equations for them.